Answer:
The water level is dropping by approximately 1.27 centimeters per minute.
Step-by-step explanation:
Please refer to the attached diagram.
The height of the conical container is 6 cm, and its radius is 1 cm.
The container is leaking water at a rate of 1 cubic centimeter per minute.
And we want to find the rate at which the water level <em>h</em> is dropping when the water height is 3 cm.
Since we are relating the water leaked to the height of the water level, we will consider the volume formula for a cone, given by:
Now, we can establish the relationship between the radius <em>r</em> and the height <em>h</em>. At any given point, we will have two similar triangles as shown below. Therefore, we can write:
Solving for <em>r</em> yields:
So, we will substitute this into our volume formula. This yields:
Now, we will differentiate both sides with respect to time <em>t</em>. Hence:
The left is simply dV/dt. We can move the coefficient from the right:
Implicitly differentiate:
Since the water is leaking at a rate of 1 cubic centimeter per minute, dV/dt=-1.
We want to find the rate at which the water level h is dropping when the height of the water is 3 cm.. So, we want to find dh/dt when h=3.
So, by substitution, we acquire:
Therefore:
Hence:
The water level is dropping at a rate of approximately 1.27 centimeters per minute.